159 



W a 



COMMUTATION 



simplification may be introduced by assuming the quantity 



to be negligibly small relatively to unity, and the final formula 

 then becomes: 



The average flux density over the zone considered is obtained 

 by dividing this quantity by 2(AB)W a sin a = rW a , whence 



The constant k has to be introduced in these formulas because 

 the flux paths are not through air only as assumed in order to 

 simplify the problem. The value of $ e will actually be greater 

 than as given by formula (72) unless k is given a value greater 

 than unity. This is due to the proximity of the pole shoes to 

 the armature where the end connections leave the slots, and also 

 to the fact that the conductors actually remain parallel to the 

 shaft for a short distance beyond the ends of the slots. The 

 numerical value of k in multipolar machines of modern design 

 will usually lie between 1.2 and 3, the latter value being taken 

 when the end connections lie on a steel or iron supporting cylin- 

 der against which they are held by bands -of steel wire. 



Should it be desired to calculate separately the average value 

 of the total e.m.f. generated in the end connections of the short- 

 circuited coil, we have 



BJL.V.T, 



10 8 

 where l e = total length of coil outside armature slots (both ends) 



in centimeters, 

 = 2\/2r, if we keep to the assumption of angle a = 45 



degrees, 



T c = number of turns in short-circuited coil, 

 and V, = speed of cutting, in centimeters per second, 



= j=? t in which D is the armature diameter in centi- 



meters, and N, is the speed in revolutions per second. The 

 factor \/2 is the necessary correction to give the component of 

 the velocity at right angles to the conductor. Inserting the 

 value of B e given by formula (73), and making the required simpli- 

 fications, the formula for voltage becomes 



